Scroll down to Fine-tune the ____ forum, where the ___ is the name of the forum you wish to make private.

Set that forum's settings to:
Require membership
Require membership
Application OR Invite only

If you go the application route, there is a setting in your Misc settings, 4th option from the bottom, that says, Access application sends email to Head Admin:. If you check this box, you will receive an email whenever someone applies.

For an invite-only forum, go into fine-tune the ___ forum as you did to create those settings, then on the left you'll see a button that says see members. Click on that, then type in username of the person you wish to let in.

You could use passwords instead of application or invite, but passwords are too easily passed around and I personally don't care for them. Still, the choice is yours to make.